Original Paper

A notion of Euler characteristic for fractals

  1. Marta Llorente1,*,
  2. Steffen Winter2,†

Article first published online: 20 DEC 2006

DOI: 10.1002/mana.200410471

Issue

Mathematische Nachrichten

Mathematische Nachrichten

Volume 280, Issue 1-2, pages 152–170, January 2007

Additional Information

How to Cite

Llorente, M. and Winter, S. (2007), A notion of Euler characteristic for fractals. Mathematische Nachrichten, 280: 152–170. doi: 10.1002/mana.200410471

Author Information

  1. 1

    Universidad Autónoma de Madrid, Facultad de Ciencias Económicas y Empresariales, Departamento de Análisis Económico: Economía Cuantitativa, Campus de Cantoblanco, Carretera de Colmenar no. 16, 28046 Madrid, Spain

  2. 2

    Mathematisches Institut, Friedrich-Schiller-Universität Jena, 07737 Jena, Germany

  1. †

    Phone: +49 3641 946186, Fax: +49 3641 946186

*Phone: +34 91 497 2877, Fax: +34 91 497 2991

Publication History

  1. Issue published online: 20 DEC 2006
  2. Article first published online: 20 DEC 2006
  3. Manuscript Accepted: 19 MAY 2005
  4. Manuscript Revised: 8 APR 2005
  5. Manuscript Received: 11 MAY 2004

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Keywords:

  • Fractal;
  • self-similar set;
  • Euler characteristic;
  • parallel set;
  • Renewal theorem;
  • convex ring

Abstract

A notion of (average) fractal Euler number for subsets of ℝd with infinite singular complexes is introduced by means of rescaled Euler numbers of infinitesimal ε -neighbourhoods. For certain classes of self-similar sets we calculate the associated Euler exponent and the (average) fractal Euler number with the help of the Renewal theorem. Examples like the Sierpinski gasket or carpet are provided. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

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